proving that a sequence is convergent : $a_n=(1+x)(1+x^2)...(1+x^n)$, $0

Question

Prove that the following sequence convergent : $a_n=(1+x)(1+x^2)...(1+x^n)$, $0<x<1$

I have proven that it is monotonically increasing, now I have to prove that it is restricted.

Answer 1

Hint. Observe that $$ \ln(a_n)=\sum_{k=1}^n\ln(1+x^k) $$ with $0<x<1$, then use
$$ \ln(1+u)\le u,\qquad u \in [0,1]. $$

Answer 2

If you just want to prove the convergence, you can argue as follows. Take the logarithm of your product (which is at least $1$)

$$0\le\log\left(\prod_{k=1}^\infty (1+x^k)\right)=\sum_{k=1}^\infty \log(1+x^k) \le \sum_{k=1}^\infty x^k\,,$$

which converges for values of $x\in(0,1)$. Then, the original product converge as well.

Answer 3

The logarithm of $a_n$ is a convergent series which is less or equal than the geometric series of $x^n$, which converges. So also $a_n$ converges